Method and system for measuring porosity of particles

ABSTRACT

A method for analyzing porosity of a particle and a medium disposed in the porosity of the particle. A video-holographic microscope is provided to analyze interference patterns produced by providing a laser source to output a collimated beam, scattering the collimated beam off a particle and interacting with an unscattered beam to generate the interference pattern for analyzation to determine the refractive index of the particle and a medium disposed in the porosity of the particle to measure porosity and the medium.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

This application claims priority to U.S. Provisional Patent Application No. 61/410,739 filed Nov. 5, 2010, which is incorporated herein by reference in its entirety.

The U.S. Government has certain rights pursuant to grants from the National Science Foundation through Grant Number DMR-0820341 and in part by the NSF through Grant Number DMR-0922680.

The present invention is directed to an improved method and system for analyzing properties of particles including particle size, indices of refraction, controlling particle porosity development and particle porosity characterization measured by holographic characterization. More particularly the invention concerns a method, system and computerized method of analysis for characterization of particle porosity by determining refractive indices of particles, such as colloidal spheres, by holographic video microscopy.

BACKGROUND OF THE INVENTION

The properties of colloidal particles synthesized by emulsion polymerization typically are characterized by methods such as light scattering, whose results reflect averages over bulk samples. Therefore even as many applications advance toward single-sphere implementations, methods for characterizing colloidal spheres typically offer only sample-averaged overviews of such properties as spheres' sizes and porosities. Moreover, such characterization methods as mercury adsorption porosimetry, nitrogen isotherm porosimetry, transmission electron microscopy and X-ray tomography require preparation steps that may affect particles' properties. Consequently, such methods do not allow for determining porosity for individual particles, particularly in suspension nor allow characterization of porosity development in particles and can even modify particle properties. Consequently, a substantial need exists for a method and system for determining particle porosity and analyzing its development in particles.

SUMMARY OF THE INVENTION

The recent introduction of holographic characterization techniques now has enabled direct characterization of the radius and refractive index of individual colloidal spheres with very high resolution. Such article-resolved measurements, in turn, provide previously unavailable information on the distribution of properties in colloidal dispersions. We have used these techniques to be able to measure the porosity of individual colloidal spheres, and to probe the processes by which porosity develops during their synthesis.

The objects, aspects, variations and features of the invention will become more apparent and have a fuller understanding of the scope of the invention described in the description hereinafter when taken in conjunction with the accompanying drawings described hereinbelow.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an in-line holographic video microscope system;

FIG. 2A illustrates a holographic video microscope image of a nominally 1.2 μm diameter polystyrene sphere in water; and FIG. 2B shows a fit of the image in 2A to the predictions of Lorenz-Mie theory for the sphere's position, {right arrow over (r_(p))}(t), its radius, a_(p), and its refractive index, n_(p); and FIG. 2C shows a distribution of measured radii and refractive indexes for a randomly selected sample of 2,500 spheres such as the example in FIG. 2A; each point represents the result for one sphere and has distribution values of gray scale level as noted in the side bar of FIG. 2C, the relative probability density ρ(a_(p),n_(p)) for finding spheres of radius a_(p) and refractive index n_(p);

FIG. 3A shows a holographic video microscope image of a nominally 1.5 μm diameter silica sphere in water; FIG. 3B shows a fit of the image in FIG. 3A to the predictions of Lorenz-Mie theory for the sphere's position, {right arrow over (r)}_(p)(t), its radius, a_(p), and its refractive index, n_(p); and FIG. 3C shows distribution of measured radii and refractive indexes for a randomly-selected sample of 1,000 similar spheres, and each point represents the result for one sphere and value levels are determined by the gray scale side bar indicator of FIG. 3C for the relative probability density ρ(a_(p),n_(p)) for finding spheres of radius a_(p) and refractive index n_(p);

FIG. 4A illustrates a distribution of droplet sizes and refractive indexes for emulsified silicone oil in water; FIG. 4B shows anti-correlated properties of a monodisperse sample of emulsion polymerized silica spheres in water; and FIG. 4C shows equivalent results for a monodisperse sample of PMMA spheres in water (note for each of these figures the gray scale side bar indicator of FIG. 3C can be used to determine value levels);

FIG. 5A illustrates distribution of the scaled volumes and porosities of individual colloidal spheres composed of polystyrene (from the data in FIG. 2C); FIG. 5B shows silica (see FIG. 4B), and FIG. 5C shows PMMA (see FIG. 4C) (note for each of these figures the gray scale side bar indicator of FIG. 3C can be used to determine value levels); and

FIG. 6A shows distribution of the scaled volumes and porosities of individual colloidal spheres composed of (a) silica (from the data in FIG. 3C; FIG. 6B shows styrene (FIGS. 3B and 3C) and PMMA (FIG. 6C), all dispersed in water; lower distributions were computed for the value of n₂ that eliminates correlations between p and V_(p)(1−p); upper plots show the distribution of scaled particle volumes for these optimal values, together with fits to Gaussian distributions (note for each of these figures the gray scale side bar indicator of FIG. 3C can be used to determine value levels).

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

The method and system includes an in-line holographic video microscope system 10 in which individual colloidal spheres are illuminated by the collimated beam 20 from a fiber-coupled diode laser 30 (iFlex Viper, λ=640 nm, 5 mW) on the stage of an otherwise conventional light microscope 40 (Nikon TE 2000U). Light 45 scattered by a sample particle 35, such as for example a sphere, interferes with the unscattered portion of the beam 20 in the focal plane of the microscope's objective lens 50 (Nikon Plan-Apo, 100×, numerical aperture 1.4, oil immersion). The preferred form of the system 10 includes eyepiece 70. The interference 55 pattern is magnified by the microscope system 10, and its intensity is recorded with a video camera 60 (NEC TI-324AII) at 30 frames/s and a resolution of 135 nm/pixel. The example in FIG. 2A shows the hologram of a polystyrene sphere dispersed in water.

Each particle's image is digitized at a nominal 8 bits/pixel intensity resolution and analyzed using predictions of the Lorenz-Mie theory of light scattering to obtain the particle's position in three dimensions, its radius, and its complex refractive index. FIG. 2B shows the pixel-by-pixel fit to the measured hologram in FIG. 1. The microscope system 10 is defocused for these measurements so that each particle's interference pattern subtends a 100×100 pixel field of view. Motion blurring is minimized by setting the camera's exposure time to 0.1 ms and the illuminating laser's intensity is regulated to make optimal use of the recording system's dynamic range. Fitting to such a large amount of data reliably yields estimates for the adjustable parameters with part-per-thousand resolution.

Hereinafter, we shall describe the method and system of the invention in the context of the particle being a sphere, although the method and system can be readily applied to any particle shape by well known modification of the Lorenz-Mie method or use of other well known analytical formalisms. The data in FIG. 2C were collected from 5,000 polystyrene spheres selected at random from a monodisperse sample (Duke Scientific, catalog number 5153, lot 26621). The suspension was diluted with deionized water so that no more than 10 spheres were in the field of view at any time, and was flowed in a microfluidic channel past the observation volume at a peak speed of 100 μm/s. The entire data set was acquired in 15 min. Each data point in FIG. 2C represents the radius and refractive index of a single sphere, with error bars comparable in size to the plot symbols. Individual symbols are shown for various gray scale levels according to the sample-estimated probability density ρ(a_(p),n_(p)) for finding a sphere with radius a_(p) and refractive index n_(p) (see side bar indicator of FIG. 3C for levels of values).

These results suggest a mean particle radius a_(p)=0.778±0.007 μm that is consistent with the manufacturer's specification. The mean refractive index n_(p)=1.572±0.003 is significantly smaller than the value of 1.5866 obtained for bulk polystyrene at the imaging wavelength. It is consistent with previous bulk measurements on colloidal polystyrene spheres.

More surprising is the distinct anti-correlation between radius and refractive index revealed by the data in FIG. 2C. Such a relationship could not have been detected with bulk probes, such as dynamic light scattering. It suggests that the larger particles in a sample are less optically dense than those on the smaller end of the distribution, and thus presumably more porous.

Each data point in FIG. 3C represents the radius and refractive index of a particular silica sphere, with error bars comparable in size to the plot symbol. The entire plot comprises results for 1,000 spheres selected at random from a monodisperse sample (Duke Scientific, catalog number 8150), with plot symbols in gray scale (see side bar for indicator of levels of values) according to the sample-estimated relative probability density; ρ(a_(p),n_(p)), for finding a sphere of radius a_(p) and refractive index n_(p). These data were obtained by flowing the aqueous dispersion through a microfluidic channel past the observation volume at a peak speed of 100 μm/2. This is slow enough that motion blurring has no measureable influence on the characterization results. The suspension was diluted with deionized water to the point that no more than 10 spheres were in the field of view at any time, thereby minimizing overlap between neighboring spheres' scattering patterns. The entire data set was acquired in 5 min.

The spheres' average radius a_(p)=0.786±0.015 μm is consistent with the manufacturer's specification. By contrast, the mean refractive index n_(p)=1.444±0.007 is significantly smaller than the value of 1.4568 for fused silica at the imaging wavelength. Similar discrepancies have been reported in previous measurements on dispersions of colloidal silica spheres.

The data in FIG. 3C also reveal a distinct anticorrelation between radius and refractive index. Such a relationship could not have been detected with bulk probes, such as dynamic light scattering. It suggests that the larger particles in a sample are less optically dense than those on the smaller end of the size distribution.

The data in FIG. 4A-4C demonstrate that observed anti-correlation is not an artifact of the technique, but rather is a common feature of colloidal samples synthesized by emulsion polymerization. FIG. 4A shows results for a very polydisperse sample of silicone oil droplets (Dow Corning 200 fluid) stabilized with Pluronic L92 surfactant in water. Although the range of particle radii is large, the distribution of refractive indexes are consistent with a bulk value of n_(p)=1.404±0.002 within the part-per-thousand resolution estimated from the uncertainty in the fitting parameters. This is reasonable because the droplets all are composed of the identical material and are not at all porous and contain fluid at bulk density. Results for smaller droplets are more strongly influenced by the surfactant, which has a bulk refractive index of 1.38. Variation in surfactant coverage thus causes variation in the apparent refractive index. The peak of the probability distribution nevertheless falls within error estimates of the refractive index of bulk silicone oil for all sizes. The lack of covariance between measured radii and refractive indexes in this sample therefore demonstrates the absence of instrumental or analytical bias in the methods used to study emulsion polymerized colloidal samples.

The data in FIGS. 4B and 4C show additional results for monodisperse aqueous dispersions of colloidal silica spheres (Duke Scientific, catalog number 8150) and colloidal polymethymethacrylate (PMMA, Bangs Laboratory, catalog number PP04N) spheres, respectively, both synthesized by emulsion polymerization. These samples both display anti-correlations between size and refractive index comparable to that of the polystyrene sample in FIGS. 2A-2C, but with different correlation coefficients. Observations on similar samples obtained from different manufacturers reveal a range of apparent correlation coefficients that may reflect differing growth conditions.

In view of the above generalization, chemically synthesized colloidal spheres are known to be less dense than the bulk material from which they are formed. The difference may take the form of voids that can be filled with other media, such as the fluid in which the spheres are dispersed. A sphere's porosity p is the fraction of its volume comprised of such pores. If the pores are distributed uniformly throughout the sphere on lengthscales smaller than the wavelength of light, their influence on the sphere's refractive index may be estimated with effective medium theory. Specifically, if the bulk material has refractive index n₁ and the pores have refractive index n₂, then the sphere's porosity is related to its effective refractive index n_(p) by the Lorentz-Lorenz relation.

$\begin{matrix} {p = \frac{{f\left( n_{p} \right)} - {f\left( n_{2} \right)}}{{f\left( n_{1} \right)} - {f\left( n_{2} \right)}}} & (1) \end{matrix}$ where f(n)=(n²−1)(n²+2). Provided that n₂ can be determined, Eqn. (1) provides a basis for measuring the porosity of individual colloidal spheres in situ.

The value of n₂ is readily obtained in two limiting cases. If the suspending medium wets the particle, then it also is likely to fill its pores. In that case, we expect n₂=n_(m), where n_(m) is refractive index of the medium. If, at the other extreme, the particle repels the solvent, then the pores might better be treated as voids with n₂=1.

We can model the growth of a colloidal sphere as the accretion of N monomers of specific volume v. Assuming a typical sphere to be comprised of a large number of monomers, and further assuming that all of the spheres in a dispersion grow under similar conditions, the probability distribution for the number of monomers in a sphere is given by the central limit theorem:

$\begin{matrix} {{{P_{N}(N)} = {\frac{1}{\sigma_{N}}\sqrt{\frac{2}{\pi}}{\exp\left( {- \frac{\left\lbrack {N - N_{0}} \right\rbrack^{2}}{2\;\sigma_{N}^{2}}} \right)}}},} & (2) \end{matrix}$ where N₀ is the mean number of monomers in a sphere and σ_(N) ² is the variance in that number.

Were each sphere to grow with optimal density, its volume would be Nv. Development of porosity p during the growth process increases the growing sphere's volume to

$\begin{matrix} {{V_{p} = {{\frac{4}{3}\pi\mspace{14mu} a_{p}^{3}} = \frac{vN}{1 - p}}},} & (3) \end{matrix}$

The probability distribution for finding a sphere of volume V therefore depends on the porosity:

$\begin{matrix} {{{P\left( V_{p} \middle| p \right)} = {\frac{1 - p}{\sigma_{V}}\sqrt{\frac{2}{\pi}}{\exp\left( {- \frac{\left\lbrack {{V_{p}\left( {1 - p} \right)} - {N_{0}v}} \right\rbrack^{2}}{2\;\sigma_{V}^{2}}} \right)}}},} & (4) \end{matrix}$ where σ_(v)=vσ_(N). An individual sphere's porosity, in turn, can be estimated from its measured refractive index through the Lorentz-Lorenz relation (Eqn. (1) above) where n₁ is the refractive index of the sphere at optimal density, n₂ is the refractive index of the surrounding fluid medium, and f(n)=(n²−1)/(n²+2). If the porosity develops uniformly as a particle grows, then the probability distribution P_(p)(p) of particle porosities will be independent of size. In that case, the joint probability P(V _(p) ,p)=P _(v)(V _(p) \p)P _(p)(p)  (5) may be factored into a term that depends only on porosity p and another that depends only on the rescaled volume V_(p)(1−p). In another form of the invention other analytical methods can be used to measure porosity, such as the “parallel model” where n_(p)=p_(n)(1−p)n₂ or the series model where 1/n_(p)=p/n₁+(1−p)n₂.

If, furthermore, a sphere's porosity develops uniformly as it grows, Eqs. (3) and (4) suggest that the rescaled volume, V_(p)(1−p), should be independent of porosity p. This is indeed the case for the data in FIG. 2C whose marked anti-correlation largely (although not completely) disappears when replotted in FIG. 5A. Here, we have used n₁=1.5866 for bulk polystyrene and n₂=1.3324 for water. Comparably good results are obtained with the silica spheres from FIG. 4B (n₁=1.4568) and the EMMA spheres from FIG. 5C (n₁=1.4887).

Small residual anti-correlations between scaled volume and porosity, particularly evident in the silica data in FIG. 5B, primarily arise in the tails of the size and porosity distribution. Considering only those spheres in the upper half of the relative probability distributions in FIGS. 5A-5C remove any statistically significant relationship as measured by Kendall's rank correlation test. The residual covariance between p and V_(p)(1−p) is less than 10−4 for the high-probability fraction in all three samples. The observed correlations in the complete data sets therefore arise primarily in the tails of the distribution, and may reflect real temporal or spatial variations in the growth conditions. The absence of correlations in the highest-probability sample is consistent with the simplified model for the development of porosity, and also with the use of effective medium theory for interpreting individual spheres' light-scattering properties. More specifically, our neglect of radial gradients in porosity appears to be justified for the samples we have investigated.

Within the assumptions of the model of Eqns. (1)-(5), the correct choice for n₂ should decorrelate the rescaled volume V_(p)(1−p) and the porosity p. We therefore select the value of n₂ for which the Pearson's correlation coefficient between p and V_(p)(1−p) vanishes. The scatter plots in FIGS. 6A-6C show the distribution of particle volumes and porosities obtained with these optimal values of n₂. The upper plots show estimates for P_(V)(V_(p)\p) obtained by integration over p, together with fits to the anticipated Gaussian form. Agreement is good enough in all three cases to justify the use of eqn (4) to interpret the experimental data.

The results for the silica sample in FIG. 6A were obtained using n₁=1.4568 for fused silica. The estimated value of n₂=1.31±0.03 is consistent with the value of 1.3324 for water at the imaging wavelength. This suggests that pores in the hydrophilic silica spheres are filled with water. The associated mean porosity, p=0.092±0.004, is comparable to the 8 percent porosity determined by low-temperature nitrogen adsorption for similar samples.

The equivalent results for the polystyrene sample in FIG. 6B were obtained using n₁=1.5866 for bulk polystyrene. In this case, the estimate value of n₂=1.13±0.05 is substantially smaller than the refractive index of either water or styrene. Rather than solvent-filled voids, the pores in the spheres seem rather to represent density fluctuations in the cross-linked polymer matrix. The failure of water to invade these pores is consistent with the hydrophobicity of polystyrene. With these choices for n₁ and n₂, the sample-averaged porosity is estimated to be p=0.054±0.008.

More surprisingly, the results for water-borne PMMA spheres plotted in FIG. 6C yield n₂=1.33±0.01, and therefore suggest that the spheres' pores are filled with water, even though PMMA is hydrophobic. The porosity, p=0.02±0.01, estimated using n₁=1.4887 for bulk PMMA, is comparable to previously reported values for similar spheres. Whereas, the polystyrene spheres appear to exclude water, the substantially less porous PMMA spheres seem to imbibe it. These observations suggest either that the two samples have substantially different pore morphologies, or else that hydrophilic groups are present within the pores of the PMMA sample.

The values obtained for single-particle porosities should be interpreted carefully and in some cases account for inhomogeneity in a particle's porosity. In some embodiments, the pores are assumed to be substantially filled with the same fluid in which the spheres are dispersed, and furthermore that the imbibed fluid retains its bulk refractive index. Departures from these assumptions can possibly give rise to some errors in the estimated porosity values. Even though single-particle values for n_(p) are believed to be both precise and accurate, the precision of the porosity distributions in the previous embodiment needs to be carefully constructed and evaluated.

Correlations in the radii and refractive indexes of colloiodal spheres measured through holographic particle characterization can be ascribed to porosity. Holographic characterization, therefore, can be used to assess the porosity of individual colloidal spheres and to gain insight into the medium filling their pores.

The present implementation uses sample averages to infer the refractive index of the medium filling the individual spheres' pores. Given this parameter, the porosity can be estimated for each sphere individually. The need to aggregate data from multiple particles could be eliminated by performing holographic characterization measurements in multiple wavelengths simultaneously. The resulting spectroscopic information, in principle, could be used to characterize both the porosity of a single sphere and also the medium filling its pores in a single snapshot.

Particle-resolved porosimetry probes the mechanisms by which porosity develops in samples of emulsion-polymerized colloidal spheres. For the samples we have studied, porosity appears to have developed uniformly as the particles grew, both within individual spheres, and throughout the sample as a whole. Differences between results for polystyrene and PMMA samples point to possible differences in the shapes or properties of their pores.

Holographic particle characterization can therefore be used to assess the porosity of individual colloidal particles and insights into the methods by which porosity develops in samples of emulsion-polymerized colloidal spheres and other particle shapes. For the variety of samples we have studied, porosity appears to develop with a probability distribution that is largely independent of the distribution of monomer number in the spheres. This leads to an apparent anti-correlation in the distribution of particles' radii and refractive indexes, which is stronger in more porous materials and is entirely absent in fully dense spheres. These observations, in turn, have ramifications for possible uses of emulsion polymerized colloidal particles in such applications as colloidal photonics.

In another aspect of the invention a conventional computer system can execute computer software stored in an appropriate memory, such as a ROM or RAM memory, embodying the analytical methodologies set forth hereinbefore to determine porosity of the subject particles.

The foregoing description of embodiments of the present invention have been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the present invention to the precise form disclosed, and modifications and variations are possible in light of the above teachings or may be acquired from practice of the present invention. The embodiments were chosen and described in order to explain the principles of the present invention and its practical application to enable one skilled in the art to utilize the present invention in various embodiments, and with various modifications, as are suited to the particular use contemplated. 

What is claimed is:
 1. A method of analyzing a particle in suspension, comprising the steps of: providing a video holographic microscope; providing a laser source for producing a collimated output beam comprised of a plurality of different wavelengths of light; scattering the collimated output beam off the particle by simultaneously using the plurality of different wavelengths of light to generate a plurality of scattered beams and a combination of the scattered beams and an unscattered portion of the output beam to generate a plurality of interference patterns; recording intensity of the interference pattern; analyzing the interference patterns to determine refractive index of the particle; and determining a measure of at least one of porosity of the particle and a medium disposed in the porosity of the particle.
 2. The method as defined in claim 1 wherein the step of analyzing the interference pattern comprises applying a Lorenz-Mie formalism to determine at least one of the refractive index of the particle and the medium disposed in the porosity of the particle.
 3. The method as defined in claim 1 further including the step of analyzing the interference pattern during growth of the particle, thereby enabling characterization of development of the porosity in the particle and the medium disposed in the porosity of the particle.
 4. The method as defined in claim 2 wherein the Lorenz-Mie formalism comprises, I(r)=|E ₀(r)+E ₀(r _(p))f _(s)(k(r−r _(p)))|², where I(r) is intensity of the interference pattern recorded at position r, E₀(r) is the electric field of the output laser at position r, r_(p) is the position of the particle, k is the wavenumber of the light, and f_(s)(kr) is the Lorenz-Mie scattering function that describes scattering of light by the particle, wherein the Lorenz-Mie scattering function depends on radius of the particle, a_(p), and on effective refractive index n_(p) of the particle.
 5. The method as defined in claim 1 further including a computer system for executing computer software to carry out the steps of analyzing the interference pattern and comparing the refractive index of the bulk to the refractive index of the particle.
 6. The method as defined in claim 4 wherein n₂ is selected from the group consisting of n_(m), refractive index of the suspension fluid wetting the particle, and n₂=1 wherein the porosity of the particle is a void.
 7. A method of analyzing a particle in suspension, comprising the steps of: providing a video holographic microscope; providing a laser source for producing a collimated output beam comprised of a plurality of different wavelengths of light; scattering the collimated output beam off the particle by simultaneously using the plurality of different wavelengths of light to generate a plurality of scattered beams and a combination of the scattered beams and an unscattered portion of the output beam to generate a plurality of interference patterns; recording intensity of the interference pattern; analyzing the interference patterns through application of a Lorenz-Mie formalism to determine at least one of the refractive index of the particle and the medium disposed in the porosity of the particle wherein the Lorenz-Mie formalism comprises, I(r)=|E ₀(r)+E ₀(r _(p))f _(s)(k(r−r _(p)))|², where I(r) is intensity of the interference pattern recorded at position r, E₀(r) is the electric field of the output laser at position r, r_(p) is the position of the particle, k is the wavenumber of the light, and f_(s)(kr) is the Lorenz-Mie scattering function that describes scattering of light by the particle, wherein the Lorenz-Mie scattering function depends on radius of the particle, a_(p), and on effective refractive index n_(p) of the particle; and determining a measure of at least one of porosity of the particle and a medium disposed in the porosity of the particle.
 8. The method as defined in claim 7 further including the step of analyzing the interference pattern during growth of the particle, thereby enabling characterization of development of the porosity in the particle and the medium disposed in the porosity of the particle.
 9. The method as defined in claim 7 further including a computer system for executing computer software to carry out the steps of analyzing the interference pattern and comparing the refractive index of the bulk to the refractive index of the particle.
 10. The method as defined in claim 7 wherein n₂ is selected from the group consisting of n_(m), refractive index of the suspension fluid wetting the particle, and n₂=1 wherein the porosity of the particle is a void. 